Consider a standard multivariate random effects model

$$\mathbf{X}_{ij} = \boldsymbol{\mu} + \boldsymbol{\alpha}_i +\boldsymbol{\varepsilon}_{ij}$$

for $i = 1,\ldots,m$ and $j = 1,\ldots,n_i$, and where $\boldsymbol{\alpha}_i \sim \mathcal{N}_p( \mathbf{0},\Sigma_B)$ and $\boldsymbol{\varepsilon}_{ij} \sim \mathcal{N}_p( \mathbf{0},\Sigma_W)$.

Then consider that $m$ samples of variable sizes $n_1, \ldots,n_m$ are taken. One can break down the total sum of squares as: $$\underbrace{\sum_{i = 1}^m \sum_{j = 1}^{n_i} (\mathbf{x}_{ij}-\bar{\bar{\mathbf{x}}})(\mathbf{x}_{ij}-\bar{\bar{\mathbf{x}}})'}_{\mathbf{SST}} = \underbrace{\sum_{i = 1}^m \sum_{j = 1}^{n_i} (\mathbf{x}_{ij}-\bar{\mathbf{x}}_{i.})(\mathbf{x}_{ij}-\bar{\mathbf{x}}_{i.})'}_{\mathbf{SSW}} + \underbrace{\vphantom{\sum_{j = 1}^{n_i} }\sum_{i = 1}^m n_i (\overline{\mathbf{x}}_{i.}-\bar{\bar{\mathbf{x}}})(\bar{\mathbf{x}}_{i.}.-\bar{\bar{\mathbf{x}}})'}_{\mathbf{SSB}}$$ Are there any results on the distribution of the $\mathbf{SSB}$ matrix - whether exact or approximations?

I have not been able to find any results with this setup, and I don't really know where to start. Any help is appreciated.



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